Rigging makehuman character in Blender 2.79
This sounds like a basic armature mis-configuration. If the armature was not provided with the model, check the following:

Vertex groups in the model are not only existent but are assigned to the right areas (use weight paint mode to check).
The model has an Armature modifier on it, and the modifier’s target is the correct armature object. What’s more, the modifier is turned on (visible in both viewport and render).
Further modifiers on the model would not be interfering with the Armature modifier results. Shrink wrap would fall into this category, while corrective smooth would not.